Question

Need help with this problem: Find the constant a, or the constants a and b, such that the function is continuous on the entire real line.

f(x) = x ^3 , x (less than or equal to) 2
ax^2 , x (greater than or equal to) 2

Answer 1

First you want to get the value of f(x) at 2 from the left side.
That would be:
\[f(x) = x^3 \Rightarrow f(2) = 2^3 = 8\]
Then you want to find the value of f(x) at 2 from from the right:
\[f(x) = ax^2 \Rightarrow f(2) = a(2)^2 = 4a\]
Now you want to see what value of a will make this continuous, so:
\[4a = 8 \Rightarrow a = 2\]

Answer 2

Okay thanks.